Fermat’s
Last Theorem
and
the
Building Blocks of the Universe
©Edward R. Close, January 18. 2018
There is a deep connection between Fermat’s Last
Theorem (FLT) and the geometric structure of the universe. It has been
overlooked by mainstream science for nearly 400 years. The connection is
simple, but has been obscured by abstract mathematical complexity and the myopic
restrictions of academic specialization. It has been 353 years since Fermat
died, and I think it is time for the importance of his work to be more fully recognized.
This essay is an attempt to do that.
There are many observations about numbers that are
easy to state and easy to understand, and yet very difficult to prove, and FLT is
a prime example of this kind of statement. Fermat said that he had found a
“marvelous” proof, but, because it was never found, and no one else could
produce one, mathematicians considered FLT nothing more than a conjecture from
1637, when Fermat made the statement, until 1994, when British mathematician
Andrew Wiles produced a 129-page proof that has been accepted by the community
of professional mathematicians, allowing FLT to finally take its place as a legitimate
mathematical theorem. Because of the complexity of Wiles’ proof, relying on
theorems that were unknown in Fermat’s day, many mathematicians doubt that
Fermat actually proved it.
Pierre de Fermat was a lawyer and judge at the
Parlement of Toulouse, in France, but his real passion was mathematics. In
1637, he wrote the following statement in Latin in the margin of his personal
copy of Arithmetica by Diophantus of Alexandria:
"Cubum autem in duos cubos, aut
quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in
infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere
cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non
caperet.”
Which, translated into English is:
"It is impossible for a cube
to be the sum of two cubes, a fourth power to be the sum of two fourth powers,
or in general for any number that is a power greater than the second to be the
sum of two like powers. I have discovered a truly marvelous demonstration of
this proposition that this margin is too narrow to contain."
The theorem stated in somewhat simpler English is:
Two whole numbers raised to any
power greater than 2, and then added together, cannot be equal to a third whole
number raised to the same power.
In simple mathematical notation:
Xn + Yn ≠ Zn if X,
Y, Z, and n are integers and n ≥3.
For ease of expression in this
discussion, I shall call the equation Xn
+ Yn = Zn Fermat’s
equation. Note that the ‘equals’ sign (=) is replaced by ≠ (is not equal to) if Fermat’s Conjecture is true. Fermat’s proofs for the case n =
3 emerged from his writings after his death, but no general proof for all n >
2 was ever found.
FLT is a statement about numbers that
is easy to state and easy to understand, but it went without proof for more
than 300 years, even though the best and brightest mathematicians tried to
prove (or disprove) it. Professional mathematicians correctly labeled Fermat’s
Last Theorem a conjecture, which is
what a mathematical statement should be called until it is either proved or
disproved. It is likely that every mathematician alive in the last three
centuries has tried to prove FLT, because the longer it went without
resolution, the more famous it became, and resolving such a puzzle would assure
the one who was first to solve it great recognition and fame; and mathematicians
rank right up there with, or very close to, physicists, as a group of people
with well-developed egos.
Karl Friedrich Gauss, arguably one
of the greatest mathematicians who ever lived, was no exception. He invented
modular algebra as a tool for use to investigate Diophantine equations, which
is what Fermat’s equation is. [Diophantus
specialized in solving equations with whole-number (integer) solutions. So,
mathematicians call equations for which only integer solutions are sought, Diophantine equations.] Gauss
proved Fermat’s Conjecture for n = 3, using modular algebra and complex numbers,
but failed to be able to generalize the proof to greater values of n. When
asked about his interest in Fermat’s Conjecture by Astronomer Wilhelm Olbers,
Gauss said:
“
I confess that Fermat's Theorem as an isolated
proposition has very little interest for me, because I could easily lay down a
multitude of such propositions, which one could neither prove nor dispose of.”
Could this disdain for FLT be the
statement of a big ego, unwilling to admit that he couldn’t prove it, even though he tried? Probably so. Even a
great man like Gauss can be blinded by his own brilliance. But I think Gauss
can be forgiven his short-sightedness, because many brilliant thinkers believe
that there are mathematical statements that can never be proved or disproved,
and that FLT is an abstract theorem with little or no practical application
outside of abstract number theory. Modern mathematicians, on the other hand, who
believe the same things, should not be let off the hook so easily, because they
have GÓ§del’s Incompleteness Theorems, which sheds some light on the questions of
proof and provability.
In 1931, Kurt GÓ§del proved, with
two decisive theorems, that no consistent system of mathematical logic is ever complete. He
proved that within any logical system, there will always be statements that,
even though they may be easy to understand, cannot be proved true or false
within the system in which they are defined. Some mathematicians thought that
FLT was actually such a statement, just as Gauss did, even though he died 76
years before GÓ§del published his proofs.
The fact is however, that GÓ§del’s
Incompleteness Theorems do not say that there are propositions that can never be proved. They only say that
there are propositions that cannot be proved using only the system of logic
within which they were stated. Every logical system is based on one or more a priori assumptions. A meaningful
statement that cannot be proved true or false within a given logical system,
may actually be provable when the a
priori assumptions of the system are either added to, or changed in some fundamental
way.
I believe that Pierre de Fermat, and Diophantus of
Alexandria, - and perhaps a few others – at least had inklings of the
importance of Fermat’s Last Theorem as it relates to the building blocks of the
universe, but modern mathematicians and physicists, and scientists in general, seem
to have missed the point, mainly because of narrow academic specialization, coupled
with a bit of human ego and pride.
With the discovery by Max Planck, already more than
100 years ago, that the mass and energy of the physical universe are meted out
in multiples of a quantum unit, it should have been obvious that if the right
units of equivalence are used, then Diophantine equations will be needed to describe
the combinations of elementary particles that make up physical reality. That,
however, has not been the case. Current mainstream science has not yet expanded
its a priori assumptions to include
quantization of the basic parameters of the physical universe. When we do so, as
we have done with the definition of Triadic Rotational Units of Equivalence
(TRUE), the true quantum units based on the physical features of the electron,
Fermat’s Last Theorem emerges as a key mathematical concept in the process of
revealing the nature of reality. Please let me explain:
The general Fermat equation, Xn + Yn = Zn with X, Y, Z, and n equal to
positive integers, is a special case of the more general quantum combination
expression:
When the basic unit of measurement is
defined as the smallest quantum equivalence unit, all cases of this expression
are Diophantine equations, and when m = 1 and n = 2, we have the equation X1
+ X2 = Z, which describes the linear actualization of the closure of
integers; for example:
1 + 2 = 3, 2 + 2 = 4, 2 + 3 = 5,…etc.
When m = 2 and n = 2, we have (X1)2
+ (X2)2 = Z2, for which all integral solutions
are quantum actualizations of Pythagorean triples; e.g., 32 + 42
= 52, 52 + 122 = 132, 72
+ 242 = 252, etc. I have derived a simple ratio formula
for producing the Pythagorean triples. See Appendix A of The Book of Atma,
Published in 1977.
When m = 3 and n = 2, we have (X1)3
+ (X2)3 = Z3, for which, Fermat’s Last Theorem
tells us there are no integer solutions. This means that there are no quantum
actualizations of this equation, because linear values cubed are volumes, and
that is why there are no combinations of two quarks forming a larger particle.
Two quarks cannot combine volumetrically to form a symmetrically stable third
particle. However, when m = 3 and n = 3, we have: (X1)3 +
(X2)3 + (X3)3 = Z3, and
we find there are quantum actualizations of this equation. For example:
33 + 43 + 53
= 63, and 13 + 63 + 83 = 93,
etc.
This is why quarks combine in threes to produce the symmetrically stable
particles known as protons and neutrons. This is just the first example of the importance
of Fermat’s Last Theorem in understanding the quantum combinations that form the
subatomic particles that make up the elements of the Periodic Table.
I proved Fermat’s Last Theorem in 1964,
327 years after Fermat’s statement, and 30 years before Wiles’ proof. I
published the original proof as an appendix to in The Book of Atma in
1977. I documented the proof and began submitting
it to professional mathematicians in 1965. For that reason, I refer to it as
FLT65 in my subsequent writings. (See posts on Proof of Fermat’s Last Theorem on
this blog and Reality Begins with Consciousness). Since 1965, I have
submitted the proof to more than fifty mathematicians, both professional and
amateur. Out of the fifty plus, four have accepted it as valid, two
professional mathematicians and two capable mathematicians with degrees in
sciences that require familiarity with advanced mathematics; but, only one has
publicly defended it. Why is this?
It must be said in passing, that the
validity of the use of FLT in the application of the Triadic Dimensional
Vortical Paradigm (TDVP), developed by Dr. Vernon Neppe and myself in 2011, to
quantum physics is completely independent of the validity or invalidity of my
1965 proof, because FLT has long been known to be true for values of n between 3
and 9, the range of TDVP. But, if FLT65 is valid, why hasn’t it been accepted
by the community of number theory mathematicians, as has Wiles’ proof? I have
published what I see as the reasons, and details of this story in other posts
on this website (search the blog archives for Fermat’s Last Theorem) but, my
intent in this post is to explain it as briefly and clearly as I can.
The FLT65 proof relies on a very simple basic
mathematical theorem known as the Division Algorithm. More specifically, it
depends on a corollary of the Division Algorithm that says that one integer,
call it A, is a factor of another integer, call that integer B, if, and only if,
the remainder when B is divided by A is zero. For example, 9/2 = 4, with a remainder
of 1, while 9/3 = 3 + 0. So, 3 is a factor of 9 but 2 is not. The requirement
for a zero remainder is patently self-evident for integers, and is it proved in
FLT65 for algebraic polynomials consisting of real numbers. This part of FLT65
is never questioned by skeptical reviewers.
In FLT65, the Fermat equation for n equal
to a prime number greater than 2 is rewritten as Zn - Yn = Xn and factored into two
polynomials, one a first-degree binomial (a polynomial of two terms, consisting
of the variable Z plus an integer
constant) and the other an (n-1)-degree polynomial with n terms:
Zn –
Yn = (Z-Y)( Zn-1 + Zn-2Y + Zn-3Y2 +•••+ Yn-1) = Xn
The factored form of the Fermat equation is
chosen so that the (n-1)-degree polynomial must be equal to the nth power of an
integer factor of X. This means that the (n-1)-degree polynomial factor must be
divisible by that integer, a factor of X. None
of this is disputable, and was not disputed by any of the reviewers.
The integer divisor, because its value is
unknown, and because the integers are closed with respect to addition, can be
represented by the variable Z minus an integer constant. When the (n-1)-degree
polynomial factor of the Fermat equation is divided by the integer represented
by Z minus a, where a is an integer constant, the remainder is a polynomial comprised of
positive integers, and thus cannot equal zero for any integer solution of the
Fermat equation. The fact that the remainder cannot equal zero for any integer
solution of the Fermat equation means that, by the Division Algorithm corollary
cited above, the (n-1)-degree polynomial cannot be divisible by an integer
factor of X, which proves FLT for all n.
Is FLT65 a valid proof of FLT, or not? It seems
that this should be a question that could be answered decisively, very quickly.
But, given the 350-year history of FLT, mathematicians consider the claim of a simple
proof an extraordinary claim, and, of course, an extraordinary claim requires extraordinary
proof.
The question is: If FLT65 is so simple that given the
Division Algorithm and its corollaries, the proof can be described in two
pages, in contrast with Wile’s proof of 129 pages, given both Ribet’s theorem
and a special case of the modality theorem for elliptical functions, why has FLT65
not been accepted by more than a handful of reviewers?
I have answered this question in the History of FLT65 and
other discussions previously posted, but the reasons can be summarized as
follows:
First, there is a cultural bias in the community of
professional mathematicians against considering the possibility that anyone
outside the academic mathematics community might be able to produce a valid
proof of FLT. This is an endemic attitude epitomized by: “If I can’t prove it,
no one can, and it’s not that important anyway.” (Reminiscent, e.g., of Gauss’
denial, and definitely reflected in Descartes’ attempts to discredit Fermat.)
I developed FLT65 while teaching secondary-school mathematics,
shortly after earning my degree in mathematics. The first professional mathematician
to whom I submitted FLT65, was a math professor at a Midwestern University, who
also happened to be the President of the state’s Academy of Sciences at the
time. He returned my proof with a brief note that said: “Your proof is invalid
because, if true, it would hold for n = 2.” Of course, the case n = 2 of the
Fermat equation is the Pythagorean Theorem equation, with integer solutions called
the Pythagorean triples, as noted above. I
was stunned. His answer was unbelievable! It revealed that he hadn’t read
the proof at all. The case n = 2 was eliminated on the first page!
Most of the university mathematics professors to whom
I submitted FLT65, simply ignored it. This is actually quite understandable,
because they receive hundreds of half-baked proofs and mathematical ramblings from
would-be mathematicians every year. Some number theory professors have form
letters they send out in response to such unsolicited proofs, while others just
refuse to waste their time reading purported proofs submitted by anyone unknown
to them.
Second, mathematicians who tried to disprove FLT65
(there were four) were not willing to go beyond trying to disprove it, probably
largely because of the endemic attitude of disbelief cited above, or because of
the fear of loosing credibility in the professional community. One of these
reviewers thought that the notation used in FLT65 was confusing, and suggested
that if the standard notations for variables and constants, integers and rational
numbers were used, the error in the reasoning would probably become clear. I rewrote
FLT65, changing the notation as appropriate, and found that it made no
difference, since for integer solutions of the Fermat equation, the only
distinction necessary was between variables and constants, which was already
done in the original FLT65, and for integer solutions, both variables and
constants are integers by definition.
Finally, my submittals of FLT65 to professional
mathematicians between 1965 and 2013, a period of nearly fifty years, were sporadic
because of my career. Working as a systems analyst, environmental engineer, and
consultant, I was involved in projects that required frequent moves from state
to state, across the country, and out of the country for prolonged periods. As
a result, some reviewers lost interest, and some. unfortunately, have passed
away. Over the years, I have kept a file of all meaningful correspondences and
attempts to disprove FLT65.
While FLT65 has failed to get support from any
mathematician with much influence in the professional mathematics community, no one has been able to actually refute it.
All the attempts to do so have involved one or more of the following
approaches: 1) The proposition that the Division Algorithm and its corollaries may
not apply to integers. 2) The production of a “counter-example”, a set of three
integers which, when substituted into the polynomials on FLT65, appear to contradict
the Division Algorithm corollary. 3) The argument that even though specific
examples failed to disprove FLT65, they could represent a loophole in the
proof.
1) The
proposition that the remainder corollary of the Division Algorithm might not
apply to integer solutions of the Fermat equation, was suggested by several
reviewers. However, none of them offered a general proof of this. In fact, they
couldn’t because the opposite is true: The corollary applies over the field of
real numbers, which includes the integers, so it applies to integer polynomials. This is stated in FLT65 and
demonstrated in the proof of the Division Algorithm and its corollaries,
included as the first part of FLT65.
2) Because
of their belief that FLT65 could not be valid, some reviewers tried to produce
counter-examples with integer values that appeared to contradict the remainder
corollary. This approach proved to be exceptionally subtle and misleading
because one can indeed find integers that, when substituted into the n = 3
Fermat equation’s second-degree (n-1) factor, will produce a value that
contains the divisor as an integer factor, even though the remainder is
non-zero. It was easy to show, however, that the integers the reviewers chose
for such examples were not solutions of the Fermat equation. For that reason, the
approach was a form of misdirection. It focused the attention on a
demonstration that had nothing to do with FLT. Not only that, if anyone could actually produce a counter-example, it would not only disprove FLT65, it would disprove Wiles' proof as well, because it would produce an integer solution for the Fermat equation.
3) One
reviewer, who appeared to be well-qualified to review FLT65, announced that he
had disproved FLT65 with a counter-example. When shown that his example was not
relevant to the Fermat equation, he admitted that his “counter-example” did not
disprove FLT65, but still maintained that it revealed a loophole in the proof,
because if integers could be found that produce a value for the polynomial
factor of the Fermat equation that contains the divisor as an integer factor,
even though the remainder is non-zero, who is to say there isn’t at least
one set of such integers that would actually
produce an integer solution to the FLT equation?
Position #3 gave me some pause, until I realized it
could only be true if proposition #1 were true, i.e., there would have to be integer
polynomials for which the Division Algorithm and its corollaries did not hold.
But, the Division Algorithm and its corollaries are proved across the field of
real numbers in the first part of FLT65, and integers are real numbers. To see
the truth of this clearly for the Fermat equation, one only has to do the
following:
Assume there is an integer solution for the Fermat
equation for some integer value of n ≥ 3, and substitute the three integers of
the solution into the factored Fermat equation, (Z-Y)( Zn-1 + Zn-2Y + Zn-3Y2 +•••+ Yn-1) = Xn.
Then the integer polynomial f(Z) =
Zn-1 + Zn-2Y + Zn-3Y2 +•••+ Yn-1 must contain
a factor of X. (In fact, it must
contain the factor raised to the nth power). And since X and Z are positive integers, Z is
larger than X, and integers are
closed with respect to addition, there is a positive integer a, such that the factor of X is equal to Z – a. Then the integer polynomial f(Z) divided by Z – a
yields a remainder equal to:
an-1 + an-2Y + an-3Y2 +•••+ Yn-1, and since a and Y are positive
integers, the remainder is non-zero for all values of a and Y. But, the
integer polynomial f(Z) can contain Z–a, if, and only if, the remainder is
zero.
If there is any lingering concern that when the
integer polynomials, f(Z) and Z–a, are reduced to single integers, A and B, respectively, (as they certainly can be, if there are integer
solutions for the Fermat equation, because integers are closed with respect to
addition and multiplication), A might
still contain B as a factor, it is dispelled
by the following demonstration:
There is no question that, if there is an integer
solution (X,Y,Z) of the Fermat
equation, the equation can be expressed as the integer polynomials displayed
above. And, as integer polynomials, Z–a
divides f(Z), if and only if the
remainder is zero. Therefore, if we set the remainder equal to zero and solve
for a, and determine the values of X, Y and Z for each value of a,
we will obtain exactly n-1 solutions for the Fermat equation. When we solve for
a, however, we find that a cannot be an integer, and therefore,
if two of the three X,Y,Z values for
any solution are integers, then the third is a non-integer. So, solving for a, produces n-1 non-integer solutions
to the Fermat equation, and one additional solution is provided by a = Z which implies X = 0, a legitimate solution of the Fermat equation. This means
that we have the n solutions of the Fermat equation, and by the Fundamental
Theorem of Algebra (FTA), there are no
more solutions.
The Fundamental Theorem of Algebra states that every
non-zero, single-variable polynomial of degree n with complex coefficients has
exactly n complex roots.
For any integral solution of the Fermat equation, f(Z)
is a non-zero, single variable polynomial of degree n, and the coefficients of
f(Z) are real numbers, and all real numbers are complex numbers with the
imaginary term equal to zero. So, there cannot be more than n solutions to the Fermat equation, and none of them are positive integer solutions with X, Y and Z equal to positive integers.
Conclusion: All of the legitimate questions raised by reviewers of FLT over the years have been eliminated and resolved. Therefore:
The FLT65 proof is complete and valid as it was written in 1965.
COMMENTARY AND TRIBUTE TO PIERRE DE FERMAT:
The FLT65 proof contains concepts that would indeed
have been available to Fermat, even though they are probably in different form
and with different notation than he would have used in 1637. I consider FLT65 to
be an elegant proof, because it relies on a deep truth about the fundamental mathematical
operation of division, which applies to all real numbers. I have also validated
the FLT65 proof in previous written presentations using the logic of infinite
descent, Fermat’s favorite method of proof. This means that Fermat definitely could
have found his “marvelous” proof.
To Pierre de Fermat I want to say:
Requiesce in pace, Pierre, tuus lumen mathematicum
esse iudicavit!
